Theorem scutfo 34011

Description: The surreal cut function is onto. (Contributed by Scott Fenton, 23-Aug-2024.)

Assertion

Ref	Expression
scutfo	⊢ |s : <<s –onto→ No

Proof of Theorem scutfo

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scutf 33933	. 2 ⊢ |s : <<s ⟶ No
2		lltropt 33983	. . . . 5 ⊢ (𝑥 ∈ No → ( L ‘𝑥) <<s ( R ‘𝑥))
3		df-br 5071	. . . . 5 ⊢ (( L ‘𝑥) <<s ( R ‘𝑥) ↔ ⟨( L ‘𝑥), ( R ‘𝑥)⟩ ∈ <<s )
4	2, 3	sylib 217	. . . 4 ⊢ (𝑥 ∈ No → ⟨( L ‘𝑥), ( R ‘𝑥)⟩ ∈ <<s )
5		lrcut 34010	. . . . 5 ⊢ (𝑥 ∈ No → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
6	5	eqcomd 2744	. . . 4 ⊢ (𝑥 ∈ No → 𝑥 = (( L ‘𝑥) |s ( R ‘𝑥)))
7		fveq2 6756	. . . . . 6 ⊢ (𝑦 = ⟨( L ‘𝑥), ( R ‘𝑥)⟩ → ( |s ‘𝑦) = ( |s ‘⟨( L ‘𝑥), ( R ‘𝑥)⟩))
8		df-ov 7258	. . . . . 6 ⊢ (( L ‘𝑥) |s ( R ‘𝑥)) = ( |s ‘⟨( L ‘𝑥), ( R ‘𝑥)⟩)
9	7, 8	eqtr4di 2797	. . . . 5 ⊢ (𝑦 = ⟨( L ‘𝑥), ( R ‘𝑥)⟩ → ( |s ‘𝑦) = (( L ‘𝑥) |s ( R ‘𝑥)))
10	9	rspceeqv 3567	. . . 4 ⊢ ((⟨( L ‘𝑥), ( R ‘𝑥)⟩ ∈ <<s ∧ 𝑥 = (( L ‘𝑥) |s ( R ‘𝑥))) → ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦))
11	4, 6, 10	syl2anc 583	. . 3 ⊢ (𝑥 ∈ No → ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦))
12	11	rgen 3073	. 2 ⊢ ∀𝑥 ∈ No ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦)
13		dffo3 6960	. 2 ⊢ ( |s : <<s –onto→ No ↔ ( |s : <<s ⟶ No ∧ ∀𝑥 ∈ No ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦)))
14	1, 12, 13	mpbir2an 707	1 ⊢ |s : <<s –onto→ No

Syntax hints:   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  ⟨cop 4564   class class class wbr 5070  ⟶wf 6414  –onto→wfo 6416  ‘cfv 6418  (class class class)co 7255   No csur 33770   <<s csslt 33902   |s cscut 33904   L cleft 33956   R cright 33957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774  df-bday 33775  df-sslt 33903  df-scut 33905  df-made 33958  df-old 33959  df-left 33961  df-right 33962

This theorem is referenced by: (None)